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Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling…
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with…
A coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the…
We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow…
By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be non-unital we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the…
Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded…
We show that with the appropriate choice of coproduct, the type B quasisymmetric functions form a Hopf algebra, and the recently introduced type B peak functions form a Hopf subalgebra.
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of…
We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and…
We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively.…
A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the…
It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We…
We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well…
In this paper, we give several necessary conditions for non-cosemisimple coalgebras being admissible. The implications simplify the classification problems for Hopf algebras of dimension 45, 105 and a few others.
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…
In this short note, we construct quasi-idempotent Rota-Baxter operators by quasi-idempotent elements and show that every finite dimensional Hopf algebra admits nontrivial Rota-Baxter algebra structures and tridendriform algebra structures.…
Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half…
Attention is focused on quantum spaces of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. Each of these quantum spaces can be…
Attention is focused on q-deformed quantum algebras with physical importance, i.e. $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry…
If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v:H\to B a morphism of right H-comodule algebras, we prove that there exists a left H-module algebra A such that B\simeq A# H. The main difference…